A005279 Numbers having divisors d, e with d < e < 2*d.

6, 12, 15, 18, 20, 24, 28, 30, 35, 36, 40, 42, 45, 48, 54, 56, 60, 63, 66, 70, 72, 75, 77, 78, 80, 84, 88, 90, 91, 96, 99, 100, 102, 104, 105, 108, 110, 112, 114, 117, 120, 126, 130, 132, 135, 138, 140, 143, 144, 150, 153, 154, 156, 160, 162, 165, 168, 170, 174, 175, 176

Offset: 1

Author

N. J. A. Sloane, Robert G. Wilson v

Keywords

• nonn

Comments

The arithmetic and harmonic means of A046793(n) and a(n) are both integers.

n is in this sequence iff n is a multiple of some term in A020886.

a(n) is also a positive integer v for which there exists a smaller positive integer u such that the contraharmonic mean (uu+vv)/(u+v) is an integer c (in fact, there are two distinct values u giving with v the same c). - Pahikkala Jussi, Dec 14 2008

A174903(a(n)) > 0; complement of A174905. - Reinhard Zumkeller, Apr 01 2010

Also numbers n such that A239657(n) > 0. - Omar E. Pol, Mar 23 2014

Erdős (1948) shows that this sequence has a natural density, so a(n) ~ k*n for some constant k. It can be shown that k < 3.03, and by numerical experiments it seems that k is around 1.8. - Charles R Greathouse IV, Apr 22 2015

Numbers k such that at least one of the parts in the symmetric representation of sigma(k) has width > 1. - Omar E. Pol, Dec 08 2016

Erdős conjectured that the asymptotic density of this sequence is 1. The numbers of terms not exceeding 10^k for k = 1, 2, ... are 1, 32, 392, 4312, 45738, 476153, 4911730, 50359766, 513682915, 5224035310, ... - Amiram Eldar, Jul 21 2020

Numbers with at least one partition into two distinct parts (s,t), sWesley Ivan Hurt, Jan 16 2022

Appears to be the set of numbers x such that there exist numbers y and z satisfying the condition (x^2+y^2)/(x^2+z^2) = (x+y)/(x+z). For example, (15^2+10^2)/(15^2+3^2) = (15+10)/(15+3), so 15 is in the sequence. - Gary Detlefs, Apr 01 2023

From Bob Andriesse, Nov 26 2023: (Start)

Rewriting (x^2+y^2) / (x^2+z^2) = (x+y) / (x+z) as (x^2+y^2) / (x+y) = (x^2+z^2) / (x+z) has the advantage that the values on both sides of the = sign in the given example become integers. A possible sequence with the name: "k's for which r = (k^2+m^2) / (k+m) can be an integer while mA053629(n) and the r's being A009003(n). If (k^2+m^2) / (k+m) = r and m satisfies the divisibility condition, then r-m also does, because (k^2 + (r-m)^2) / (k + (r-m)) = r as well, confirming Pahikkala Jussi's comment about the existence of two distinct values for his u.

The fact that 15 is in the sequence is not so much because (15^2 + 10^2) / (15^2 + 3^2) = 1.3888... = (15+10) / (15+3), as indicated by Gary Detlefs, but rather because (15+10) | (15^2 + 10^2). And since r = (15^2 + 10^2) / (15+10) = 13, the second value that satisfies the divisibility condition is 13-10 = 3, so (15^2 + 3^2) / (15+3) = 13 as well.

Since (k+m)| (k^2 + m^2) is equivalent to (k+m) | 2*k^2 as well as to (k+m) | 2*m^2, both of these alternative divisibility conditions can be used to generate the same sequence too. (End)

References

• R. K. Guy, Unsolved Problems in Number Theory, E3.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
• Paul Erdős, On the density of some sequences of numbers, Bull. Amer. Math. Soc. 54 (1948), 685--692 MR10,105b; Zbl 32,13 (see Theorem 3).
• Paul Erdős, Some unconventional problems in number theory, Journées Arithmétiques de Luminy, Astérisque 61 (1979), p. 73-82.
• Paul Erdős, Some unconventional problems in number theory, Mathematics Magazine, Vol. 52, No. 2 (1979), pp. 67-70.
• Paul Erdős, On some applications of probability to analysis and number theory, J. London Math. Soc., Vol. 39, No. 1 (1964), pp. 692-696, alternative link.
• Planet Math, Integer Contraharmonic Means, Proposition 4.
• Planet Math, Contraharmonic proportion
• Robert G. Wilson v, Letter, N.D.

Crossrefs

Subsequence of A024619 and hence of A002808.

Cf. A010814, A089341, A020886, A046793, A174903, A174905, A237271, A237593, A239657.

Programs

Haskell

a005279 n = a005279_list !! (n-1)
a005279_list = filter ((> 0) . a174903) [1..]
-- Reinhard Zumkeller, Sep 29 2014

Maple

isA005279 := proc(n) local divs,d,e ; divs := numtheory[divisors](n) ; for d from 1 to nops(divs)-1 do for e from d+1 to nops(divs) do if divs[e] < 2*divs[d] then RETURN(true) ; fi ; od: od: RETURN(false) : end; for n from 3 to 300 do if isA005279(n) then printf("%d,",n) ; fi ; od : # R. J. Mathar, Jun 08 2006

Mathematica

aQ[n_] := Select[Partition[Divisors[n], 2, 1], #[[2]] < 2 #[[1]] &] != {}; Select[Range[178], aQ] (* Jayanta Basu, Jun 28 2013 *)

PARI

is(n)=my(d=divisors(n));for(i=3,#d,if(d[i]<2*d[i-1],return(1)));0 \\ Charles R Greathouse IV, Apr 22 2015

Python

from sympy import divisors
def is_A005279(n): D=divisors(n)[1:]; return any(e<2*d  for d,e in zip(D, D[1:]))
# M. F. Hasler, Mar 20 2025

Formula

a(n) = A010814(n)/2. - Omar E. Pol, Dec 04 2016